As U.S. President Barack Obama touched down in China on Sunday, his visit marked a pivotal moment in the two countries' relationship. While the U.S. and China once met to discuss topics of mutual importance, their talks are now occupied with issues of significance to the entire planet. From North Korea to Iran, global warming to global trade imbalances, Washington is increasingly dependent on Beijing's cooperation. The U.S. and China often find things to disagree about. As the world's most powerful democracy and the world's most powerful authoritarian state, they are bound to clash, even as their economic relationship draws them closer. But at a time when so much rests on their relationship, it's important also to consider where they are (mostly) in agreement. Here are five key areas:

Trade
Trade between the U.S. and China has been a heated subject in recent months. After Obama imposed tariffs in September of up to 35% on Chinese-made tires to protect U.S. jobs due to a surge in Chinese imports, China retaliated in October with new levies on nylon imports. This month, the U.S. slapped duties of up to 99% on some Chinese-made steel pipes. China announced soon after that it was looking into imports of U.S.-made cars from manufacturers that received government support. The trend has economists worried about a trade war. But U.S. officials dismiss that notion, arguing that the affected goods comprise a small part of the massive trade relationship that surpassed $400 billion last year. The global economic slump has no doubt exacerbated tensions, but the U.S. and China have matured in how they discuss their trade differences. "They're working through a lot of scattered issues, but they are working through the WTO," says James McGregor, the former chairman of the American Chamber of Commerce in China. "In the old days, every trade issue would become a very public and unstructured argument." (Read "Obama in Southeast Asia: Mending Fences in a Key Region.")

Green Technology
In an Oct. 23 speech at the Massachusetts Institute of Technology, Obama said the country that wins the race to develop renewable forms of energy "will be the nation that leads the global economy." That's something China's leadership heartily agrees with. China has become the world's leading producer of greenhouse gases, and many of its big cities choke on smog from cars and coal-fired power plants. But it is also a global pioneer in renewable energy. The government has mandated that by next year 3% of its power must come from renewable sources, excluding hydroelectricity, in which it is already among the world's top producers. That figure jumps to 8% for 2020. "The top leadership, they are all engineers," says Julian Wong, an analyst with the Center for American Progress, a liberal think tank. "They look at how the U.S. has grown by being a technological leader. China wants to do the same. They've seen the low-carbon sector as one way to do it."

North Korea
Nearly 60 years ago the U.S. and China went to war with each other on the Korean Peninsula. Now the two countries are closer than ever in their attitude toward the despotic regime. Both the U.S. and China desire an end to the North's nuclear-weapons program. Beijing has hosted the six-party talks aimed at finding a peaceful resolution, and last month it lobbied Pyongyang to return to the bargaining table. During a visit to Beijing, U.S. Assistant Secretary of State Kurt Campbell said, "I have rarely seen better coordination between China and the United States in particular. There is a virtually unprecedented level of acceptance of basic goals and ambitions associated with the six-party talks and negotiations with North Korea." Of course, China's cooperation has limits. It remains the North's closest ally. While the U.S., especially under the Bush Administration, was willing to confront and isolate Kim Jong II, Beijing doesn't want to push the North to the point of collapse. It fears the repercussions of a failed North Korea as much as, or perhaps more than, it does a nuclear-armed dictator. (See rare pictures from inside North Korea.)

Mutual Uncertainty
In the 1950s, columnist Walter Winchell proposed calling the Russians "frenemies" of the U.S. Last year, comedian Stephen Colbert suggested frenemy as a term for China. In fact, Americans and Chinese agree that they aren't sure what to think of each other. According to a poll this month by Thompson Reuters/Ipsos, 34% of American respondents said China was the country with which the U.S. had the most important bilateral relationship, ahead of Britain and Canada. But 56% categorized China as an adversary and just 33% called it an ally. That ambivalence is reflected on the other side of the Pacific. While Obama is popular in China, his celebrity has done little to move public opinion about the U.S. overall, according to the Pew Global Attitudes Project. The favorability rating of the U.S. among Chinese is just 47%, exactly what it was two years ago.

Education
While the U.S. and China aren't certain what to think of one another, each country has taken an interest in studying its counterpart. The U.S. has long been the destination of choice for Chinese college students, but China has not enjoyed the same prominence for young Americans. That's changing. More than 11,000 Americans studied abroad in China last year, a 25% increase over the previous year, making it the fifth most popular destination, according to the Institute of International Education. Students from China are already the second largest group of foreign students at U.S. universities, after those from India, but their numbers are increasing as well. Last year, 81,000 mainlanders studied in the U.S., a 20% increase from 2008. Growing educational exchanges can only help relations, says Zhu Feng, a professor of international studies at Peking University. "I think it will be an indispensable parallel to economic cooperation and political cooperation," he says. "It's the right time for both sides to take a more serious look at each other. Part of that is for ordinary people to understand what the U.S. is and what China is."